Your scenario would be called out by median household income, or better median disposable household income. Even the good old GDP per capita covers your case.
Workforce participation also can be valuable instead of or in addition to unemployment numbers, since you fall out of the count once unemployment benefits expire. However, we need to look at it by age bracket. Lower workforce participation between 20 and 60 is probably bad whereas higher workforce participation over 60 might also be bad.
IMO the problem isn't that the metrics aren't there but that the public discourse either lacks motivation, understanding or incentive to take a proper look. That every discussion of these numbers on social media has a substantial portion of people not understand the difference between median and mean certainly doesn't give me confidence this will ever improve.